A hole had just appeared in the Galaxy . ..... species, an event that we call speciation. ...... conference on Resaerch in computational molecular biology, pages ...
Using Trees to Capture Reticulate Evolution
Lateral Gene Transfers and Cancer Progression
ALI TOFIGH
Doctoral Thesis Stockholm, Sweden, 2009
TRITA-CSC-A 2009:10 ISSN-1653-5723 ISRN–KTH/CSC/A–09/10-SE ISBN 978-91-7415-349-1
KTH School of Computer Science and Communication SE-100 44 Stockholm SWEDEN
Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges till offentlig granskning för avläggande av teknologie doktorsexamen i datalogi fredagen den 12 juni 2009 klockan 10.00 i Svedbergssalen, Albanova, Roslagstullsbacken 21, Stockholm. © Ali Tofigh, 2009 Tryck: Universitetsservice US AB
To Bita
v Abstract The historic relationship of species and genes are traditionally depicted using trees. However, not all evolutionary histories are adequately captured by bifurcating processes and an increasing amount of research is devoted towards using networks or network-like structures to capture evolutionary history. Lateral gene transfer (LGT) is a previously controversial mechanism responsible for non tree-like evolutionary histories, and is today accepted as a major force of evolution, particularly in the prokaryotic domain. In this thesis, we present models of gene evolution incorporating both LGTs and duplications, together with efficient computational methods for various inference problems. Specifically, we define a biologically sound combinatorial model for reconciliation of species and gene trees that facilitates simultaneous consideration of duplications and LGTs. We prove that finding most parsimonious reconciliations is NP-hard, but that the problem can be solved efficiently if reconciliations are not required to be acyclic—a condition that is satisfied when analyzing most real-world datasets. We also provide a polynomial-time algorithm for parametric tree reconciliation, a problem analogous to parametric sequence alignment, that enables us to study the entire space of optimal reconciliations under all possible cost schemes. Going beyond combinatorial models, we define the first probabilistic model of gene evolution incorporating a birth-death process generating duplications, LGTs, and losses, together with a relaxed molecular clock model of sequence evolution. Algorithms based on Markov chain Monte Carlo (MCMC) techniques, methods from numerical analysis, and dynamic programming are presented for various probability and parameter inference problems. Finally, we develop methods for analysis of cancer progression, a biological process with many similarities to the process of evolution. Cancer progresses by accumulation of harmful genetic aberrations whose patterns of emergence are graph-like. We develop a model of cancer progression based on trees, and mixtures thereof, that admits an efficient structural EM algorithm for finding Maximum Likelihood (ML) solutions from available cross-sectional data.
Contents Abstract
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Contents
vii
List of Publications
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Acknowledgments
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1 Introduction
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2 Evolution and Describing its History 2.1 Introduction to Genetics and Genomics . . . . . . . . 2.2 Genome Evolution . . . . . . . . . . . . . . . . . . . . 2.3 Speciations and Organismal Trees . . . . . . . . . . . . 2.4 Phylogeny and Tree Reconstruction . . . . . . . . . . . 2.5 Gene Duplications and Lateral Gene Transfers . . . . 2.6 Progression in Cancer—an Evolutionary Phenomenon
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3 Computational Techniques 3.1 Ockham’s Razor and Parsimony . . . . . . . . 3.2 Parameterized Complexity . . . . . . . . . . . 3.3 Maximum Likelihood Estimation with EM . . 3.4 Bayesian Inference with Markov Chain Monte
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4 Computational Methods and Models for Duplications and LGTs 4.1 Trees Within Trees . . . . . . . . . . . . 4.2 The Duplication-Loss Model . . . . . . . 4.3 The Transfer-Loss Model . . . . . . . . 4.4 The Duplication-Transfer-Loss Model . . 4.5 DTL-scenarios . . . . . . . . . . . . . . 4.6 A Comprehensive Probabilistic Model of vii
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Contents
5 Modeling Cancer Progression 51 5.1 Overview of Current Methods . . . . . . . . . . . . . . . . . . . . . . 51 5.2 Hidden-variable Oncogenetic Trees . . . . . . . . . . . . . . . . . . . 52 6 Overview of Included Articles and Manuscripts
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Bibliography
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List of Publications • Simultaneous Identification of Duplications and Lateral Gene Transfers A. Tofigh, M. Hallett, and J. Lagergren Based on [67], which was presented at RECOMB 2004 submitted • Inferring Duplications and Lateral Gene Transfers— An Algorithm for Parametric Tree Reconciliation A. Tofigh and J. Lagergren Manuscript • Detecting LGTs Using a Novel Probabilistic Model Integrating Duplication, LGTs, Losses, Rate Variation, and Sequence Evolution A. Tofigh, J. Sjöstrand, B. Sennblad, L. Arvestad, and J. Lagergren A. Tofigh and J. Sjöstrand have contributed equally to this manuscript Manuscript • A Global Structural EM algorithm for a Model of Cancer Progression A. Tofigh and J. Lagergren Manuscript
1
Acknowledgments The journey has been long and laborious, but nonetheless quite memorable. I am grateful to all those who provided support and made the tough times less burdensome, and special thanks go to the people who contributed, both directly and indirectly, to the work presented in this thesis. Foremost among the latter is my supervisor Jens Lagergren. He is truly one of the coolest professors I have met and a true scientist—never stressed about research, always stressed about administration and teaching duties. He is also one of the fastest thinkers I know. It is only through years of training provided by myself and other students that he has learned to come down to the level of us mere mortals when explaining his scientific ideas. I’m especially grateful for his constant support and calm attitude in the face of impending disasters! Others who have made contributions to this thesis include Joel, who has worked hard during weekends to finish experiments and graphs, and Lasse and Bengt with whom I’ve had many scientific discussions. During the years as a PhD student I have shared office space with some very interesting personalities. Örjan, my brother in arms, who managed to write his thesis and become a first-time father in a matter of just a few months. A true Swedish hero! Johannes, is there anyone you don’t know or haven’t met? Isaac, it takes sharp minds and brainy people like yourself to build a search engine whose name is now an official verb. Marcus, we never shared office space, but it feels like we did. Finally, there was Samuel, who left Stockholm and never looked back. I’m also glad that I got to meet Jens’s new PhD students, Joel and Hossein. Somehow, the blend of Hossein’s evilness, Joel’s enthusaism, and my lust for revenge led to one of the most successful practical jokes at our department (sorry Jens, but your reaction was priceless!). A source of welcome distraction has been the (ir)regular poker nights, a tradition I hope will continue for a long time to come. In short, it has been a pleasure losing to you all: Håkan, Marcus, Katarina, Per, Pär, Diana, Anna, Andreas, Kristoffer, Jenny, Tomas, Björn, Aron, Maria, Erik Sj. Finally, there is the one person without whom I would not have survived through it all, and who has had to put up with my shifting moods, especially towards the end: my lovely wife Bita, you are too good to be true!
3
There is grandeur in this view of life, with its several powers, having been originally breathed into a few forms or into one; and that, whilst this planet has gone cycling on according to the fixed law of gravity, from so simple a beginning endless forms most beautiful and most wonderful have been, and are being, evolved. — Charles Darwin, On the Origin of Species
Chapter 1
Introduction Darwin’s observations during his five-year voyage on HMS Beagle laid the foundation for the body of work that he would later produce, providing compelling evidence for a process of descent with modification and natural selection. Although evolutionary ideas had been formulated in various forms before Darwin, his work popularized the idea among scientists as well as the general public. Of course, Darwin could not have known that long strands of DNA molecules, and the genes located on them, are the vehicles on which traits are inherited from generation to generation, or that discrete events such as recombination, mutation, duplication, and gene transfer are responsible for modification. Much of Darwin’s observations were based on morphological similarities and dissimilarities between species. In fact, not long ago, classification of species and inference of their evolutionary histories were mainly based on morphological data. With the emergence of modern molecular biology, we have at our disposal a detailed model of the mechanisms of inheritance, and so the study of evolution has largely shifted towards the study of DNA and genes. Methods for the construction of gene trees showing the relationship among homologous genes have been particularly useful. Trees or networks showing the evolutionary history of species are often inferred from sets of gene trees. This problem would be trivial if the history of genes simply followed that of species (or vice versa!), but genome content, and with it, the history of genes, is constantly changing in ways that do not always reflect the history of species. Prominent among these events is gene duplication, which has been extensively studied both biologically and computationally. Another event, one that has risen to prominence more recently, is lateral gene transfer (LGT), also known as horizontal gene transfer (HGT). Many computational methods have been developed that incorporate either duplications or LGTs, but few have attempted to incorporate both. A major part of this thesis concerns the development of computational models and methods for the simultaneous inference of duplications and LGTs. An evolutionary process on a much smaller scale is seen in cancer progression.
7
8
CHAPTER 1. INTRODUCTION
Cancer progresses via accumulation of genetic changes, and evolutionary mechanisms such as selection, competition, predation, and genetic drift characterize this process. The order in which different genetic changes appear during progression of the disease varies between distinct types of cancer. Although cancer progression is best described using graphs or networks, construction of such networks from available data is quite difficult. The last part of this thesis is concerned with a model of cancer progression based on trees, and mixtures thereof, for which we are able to develop efficient algorithms. The outline of the rest of this thesis is as follows. Chapter 2 provides biological background on evolution and cancer progression. Also, a discussion on the use of trees and methods for tree reconstruction is provided. In Chapter 3, we introduce the major computational techniques that our methods are based on. Chapter 4 describes problems and methods for inference of duplications and lateral gene transfers. Section 4.5 describes the combinatorial model of gene evolution presented in Papers I and II, and Section 4.6 describes the probabilistic model of gene evolution presented in Paper III. Finally, Chapter 5 describes the methods previously developed for construction of cancer progression pathways, and the model and algorithms presented in Paper IV are discussed in Section 5.2. For a brief description of the articles included in this thesis, see Chapter 6.
9
A hole had just appeared in the Galaxy . . . Somewhere in the deeply remote past it seriously traumatized a small random group of atoms drifting through the empty sterility of space and made them cling together in the most extraordinarily unlikely patterns. These patterns quickly learnt to copy themselves (this was part of what was so extraordinary about the patterns) and went on to cause massive trouble on every planet they drifted on to. That was how life began in the Universe. — Douglas Adams, The Hitchhiker’s Guide to the Galaxy
Chapter 2
Evolution and Describing its History The underlying theme of the work presented in this thesis is molecular evolution and computational methods for its study. In this chapter we will give a brief background on and overview of molecular evolution in general, followed by a more detailed discussion of the evolutionary events at the focus of this thesis. The outline of this chapter is as follows. Sections 2.1 and 2.2 give a brief overview of molecular evolution. In Section 2.5, we will discuss two important evolutionary events that have made a substantial impact in the genetic composition of organisms and which constitute a major focus of this thesis, namely, gene duplications and lateral gene transfers. The process of speciation, i.e., the evolutionary process in which new species emerge from existing ones, is discussed in Section 2.3. Trees have long been used as tools to depict the evolutionary history of organisms. Today they are also used to depict the history of genes that share a common ancestry, so-called homologous genes. A brief discussion of the use of trees and methods for their construction is given in Section 2.4. A seemingly different, yet closely related, subject is that of cancer progression. “Cancer” is a name that refers to a large class of diseases with uncontrolled cell growth and proliferation as a common characteristic. The abnormal properties of cancerous cells are due to accumulation of harmful genetic changes. This process, called somatic evolution, is very similar to evolution of species and is discussed in Section 2.6.
2.1
Introduction to Genetics and Genomics
The observation that offspring inherit traits from their parents has long been used by humans, e.g., in breeding of animals and plants. Gregor Mendel performed the first systematic study of the basis of inheritance for some simple discrete traits, such as the color of the flower of the common pea plant [107, 108]. His discoveries 11
12
CHAPTER 2. EVOLUTION AND DESCRIBING ITS HISTORY
concerning dominant and recessive traits became known as Mendelian inheritance. Mendel’s work did not receive any significant attention until it was rediscovered in the beginning of the 20th century. Research then led to the discovery that genes, the basic functional units of heredity, reside on the chromosomes, a discovery that was awarded with the Nobel prize in 1933. Although chromosomes were identified as the carriers of genetic material, the composition of the genetic material was yet unknown. The first experiments showing that the genetic information was contained in the DNA of chromosomes were performed by Avery et al. [8] and was later confirmed by Hershey and Chase [77]. James D. Watson and Francis Crick published the first accurate model of DNA structure in 1953 [150], and the genetic code was cracked by Har Gobind Khorana, Robert W. Holley, and Marshall Nirenberg, who shared the Nobel prize in physiology in 1968. The central dogma of molecular biology, that the flow of information in the cell goes from DNA to mRNA, to protein, but never from protein to nucleic acid, was formulated by Francis Crick [56, 31]. Today, technological advances have enabled us to sequence entire genomes. All 6 billion bases comprising the genome of James D. Watson were sequenced in two months time in 2008 [152]. One of the challenges of the future lies in constructing computational tools for extracting functional information from sequence data. In the following, we give a brief overview of the molecular machinery of the cell responsible for reading the genome and producing the proteins that are responsible for most of a cell’s functions. For more in-depth information, we refer to the standard text book by Bruce Alberts et al. [1].
The Central Dogma of Molecular Biology The cell is the smallest structural and functional unit of an organism that is classified as living. There are two types of cells: eukaryotes comprising multicellular animals, plants, fungi, as well as unicellular organisms, and prokaryotes such as bacteria. “Karyose” comes from a Greek word meaning kernel, “pro” means before, and “eu” means true. So prokaryotic means “before a nucleus”, and eukaryotic means “possessing a true nucleus”. The name emphasizes the fact that eukaryotes carry their genetic material inside a cell nucleus, while prokaryotes have no such compartment and the genetic material is held within the cytosol. The genetic material of both eukaryotes and prokaryotes consists of long molecules of deoxyribonucleic acid (DNA). As the eukaryotic DNA molecules are very long and have to fit in a small nucleus, they are folded up into chromosomes in a highly organized manner. The prokaryotic DNA, on the other hand, is present as circular naked DNA molecules. DNA acts like an instruction manual and its sequence provides all the information needed for a cell to function. The information is first copied to ribonucleic acid (RNA) before being transformed into proteins—this is the so-called dogma of molecular biology, namely that information passes from nucleotides to amino acids but never in the opposite direction. The functional units in the DNA
2.1. INTRODUCTION TO GENETICS AND GENOMICS
13 Purines
pentose Base glycosidic bond
Adenine
OH = ribose H = deoxyribose
Guanine
Pyrimidines
nucleoside nucleotide monophosphate nucleotide diphosphate nucleotide triphosphate
Figure 2.1:
R
Cytosine
Uracil
Thymine
The structure of nucleotides.
that code for RNA or proteins are called genes. Each gene encodes one or a set of similar proteins, and each protein performs a specialized function in the cell. Cells use the two-step process of transcription and translation to read genes and produce the strings of amino acids that make up a protein. The production of the various proteins is one of the most important processes occurring inside a cell as proteins not only form structural components of the cell, but they also compose the enzymes that catalyze the production of other organic biomolecules required for the cell to function.
DNA and RNA Structure DNA is the carrier of genetic information composed of four different nucleotides. Each nucleotide is composed of three parts: (1) a nitrogenous base known as purine (adenine (A) and guanine (G)) or pyrimidine (cytosine (C) and thymine (T)); (2) a sugar, deoxyribose; and (3) a phosphate group. The nitrogenous base determines the identity of the nucleotide, and individual nucleotides are often referred to by their base (A, C, G, or T), see Figure 2.1. One DNA strand can consist of up to several hundred million nucleotides. The nucleotide T can form a hydrogen bond with A, and C with G, making a double-helix formed by two anti-parallel complementary strands, see Figure 2.2. RNA is very similar to DNA, the only difference being that the pyrimidine base thymine is replaced by uracil (U) and the ribose comes in its fully hydroxylated form. Together, the presence of uracil in place of thymine, and the 2 -OH in the ribose constitute the two chemical differences between RNA and DNA. Also, RNA does not form a double helical structure and is in general single-stranded. There are many types of RNA present in the cell distinguished by their functional role. Three of these, namely messenger RNA (mRNA), transfer RNA (tRNA), and ribosomal RNA (rRNA), will be discussed in further detail below.
14
CHAPTER 2. EVOLUTION AND DESCRIBING ITS HISTORY
Figure 2.2:
DNA double helix.
Transcription Transcription refers to the transfer of the genetic code from DNA to a complementary RNA and occurs in the cell nucleus. The mRNA serves as an intermediate between DNA and protein. The transcription starts with the enzyme RNA polymerase attaching and unzipping the DNA molecule into two separate strands after which it binds to the promoter segment of DNA that indicates the beginning of the single strand of DNA to be copied. It moves along the DNA and matches each DNA nucleotide with a complementary RNA nucleotide to create a new RNA molecule patterned after the DNA. The copying of the DNA continues until RNA polymerase reaches a termination signal, i.e., a specific set of nucleotides that mark the end of the gene to be copied. When the RNA polymerase has finished copying a particular segment of DNA, the DNA reconfigures into the original double-helix structure. In prokaryotes, this RNA needs no further processing and provides the blueprint which directs protein synthesis. However, in eukaryotes, this RNA strand (the transcript) is first processed into mature mRNA. The processing involves the removal of intervening non-coding sequences, so-called introns. The newly created mRNA is then exported out of the nucleus and into the cytoplasm where translation can take place.
Translation Translation refers to the process of converting the information contained in an mRNA molecule into a sequence of amino acids that bind together to form a protein. In the cytosol, mRNA molecules bind to protein-RNA complexes called ribosomes. Each ribosome includes a large and a small subunit containing rRNA and more than 50 proteins. The small and large subunits of the ribosome surround the mRNA after which tRNA molecules carrying amino acids attach to the ribosome and mRNA to create the polypeptide chain, see Figure 2.3. There are several types of tRNA molecules each, containing a unique three base region called the anticodon that can base pair to the corresponding three base codon on the mRNA. Each type of tRNA
2.2. GENOME EVOLUTION
Figure 2.3:
15
A schematic view of translation.
molecule can only carry one unique amino acid, but may base pair to more than one codon sequence on the mRNA. Hence, each three base codon signals for the inclusion of a specific amino acid, but the same amino acid can be coded by several different codons. The correspondence between codons and amino acids is called the genetic code and is shown in Figure 2.4.
2.2
Genome Evolution
About 3.5 billion years ago, cells similar to modern day bacteria had appeared. There is evidence for the existence of eukaryotic cells 1.4 billion years ago with the first multicellular animals appearing around 640 million years ago [21]. In this section, we provide a few examples of the diverse (molecular) evolutionary events that have shaped present day genomes during millions of years of evolution. Although cells have acquired highly complex and accurate mechanisms for DNA replication and repair, a cell can still fail to create exact copies of its chromosomal DNA during cell division. In fact, such failures are the predominant causes of genetic changes during evolution, although transposable DNA elements also play a major role. In the context of this thesis we assume that there exists a reference genome that is representative of the genome of all individuals belonging to a certain species. For now, we also assume that the concept of species and the classification of individuals as belonging to one species is unproblematic, though as we will see, this has been contested in recent years mainly withing the prokaryotic domain. We can imagine following the fate of a single gene as it is passed from one gen-
CHAPTER 2. EVOLUTION AND DESCRIBING ITS HISTORY
16
C
A
G
Figure 2.4:
2nd Position C A Ser Tyr Ser Tyr Ser * Ser * Pro His Pro His Pro Gln Pro Gln Thr Asn Thr Asn Thr Lys Thr Lys Ala Asp Ala Asp Ala Glu Ala Glu
G Cys Cys * Trp Arg Arg Arg Arg Ser Ser Arg Arg Gly Gly Gly Gly
U C A G U C A G U C A G U C A G
3rd Position
1st Position
U
U Phe Phe Leu Leu Leu Leu Leu Leu Ile Ile Ile Met Val Val Val Val
The genetic code. A * indicates stop-codon.
eration to the next in one species. For various reasons, we would observe changes to the DNA sequence of the gene. On a small scale, we would detect substitutions, insertions, and deletions involving a few nucleotides. Translocations, exon duplications, exon shuffling, and gene conversion are examples of other small scale events that alter the sequence of a gene. Mathematical models of sequence evolution, so-called substitution models, have been proposed and are routinely used, e.g., to reconstruct trees showing the relationship between genes from different organisms. These models together with tree reconstruction methods are discussed in Section 2.4. Taking genes to be atomic units, i.e., ignoring changes to the DNA sequence, we would see larger scale events that affect entire genes. For example, portions of chromosomes are sometimes duplicated or deleted and may affect a whole set of genes that reside on those segments. We would see how genes are lost, for example due to segmental deletions that remove the sequence or a part of the sequence from the genome altogether, or deleterious mutations that cause the silencing of the gene. The number of genes can also increase via events such as gene duplications (caused, e.g., by segmental duplications or reverse transcription), lateral gene transfers (the transfer of genetic material from one species or individual to another), or interstitial deletions (segmental deletions that do not include chromosomal endpoints; these may cause two genes to be fused together into one gene). On the largest scale, we have whole genome duplications that doubles the number of chromosomes and
2.3. SPECIATIONS AND ORGANISMAL TREES
17
hence also the number of genes in a species, but is usually followed by massive gene losses. The work presented in this thesis deals mainly with two of the evolutionary events mentioned above, namely gene duplications and lateral gene transfers. These are therefore discussed in more detail later in Section 2.5.
2.3
Speciations and Organismal Trees
Ever since Darwin’s work popularized the idea of evolution, trees have been widely used to depict the historic relationship between species. When groups of individuals belonging to the same species are isolated from each other, they are independently affected by the evolutionary processes. Over time the groups will form distinct species, an event that we call speciation. Trees are often the best representation of the process of speciation in higher organism, although plants and fish are known to hybridize to form new species. In these cases, the speciation process is best represented by networks. The classification of micro-organisms into species can sometimes be problematic. For example, it has become increasingly apparent that lateral gene transfers have played a major role in prokaryotic evolution, and some have argued that a species tree representing prokaryotic evolution, at least the evolution of certain groups of taxa, may not exist. Others argue that although lateral gene transfers have played a major role, the notion of species and species trees are still meaningful representations of the evolutionary history of prokaryotes.
2.4
Phylogeny and Tree Reconstruction
Before the emergence of the modern theory of molecular evolution, classification of species and inference of phylogenies were based on morphological data. With the modern understanding of the molecular mechanisms of inheritance, practice has shifted to using DNA or amino acid sequence data as the basis for reconstructing evolutionary histories. The history of a set of homologous genes is usually adequately represented by a tree, although there are some events such as gene conversion and recombination that are responsible for creating non tree-like histories. The history of species, whether represented by trees or networks, is often inferred from a set of gene trees. In this section we will discuss some of the more popular methods for gene tree reconstruction. Computational methods for tree reconstruction attempt to find a tree or a set of trees that are optimal according to some criteria. Several different criteria have been used when devising computational methods for tree reconstruction. A number of methods have been proposed based on parsimony where we seek the tree that requires a minimum number of evolutionary events to explain the data (see Section 3.1 for a general discussion of parsimony). The first to suggest the use of parsimony as a criterion for tree reconstruction were Edwards and Cavalli-Sforza [45].
CHAPTER 2. EVOLUTION AND DESCRIBING ITS HISTORY
18
AATCC AAGCC CATCG CAACG CCTCC
S1 S2 S3 S4 S5
AATCC
(a)
AAGCC
CATCG
CAACG
CCTCC
(b)
CATCG
AATCC
0
1 0
2
CATCG
0
1 0
AATCC
AATCC
0
CATCG
1
AAGCC
1
CATCG (c)
CAACG
AATCC 2
CCTCC
0
AATCC
CATCC 1
CATCG 1
AAGCC
0
CATCG
1
CAACG
CCTCC
(d)
Figure 2.5:
Maximum parsimony example. (a) shows a set of five DNA sequences for which we seek a parsimony tree. (b) shows an arbitrary tree whose leaves are associated with the sequences. (c) shows one way to assign hypothetical sequences to the internal vertices to the tree in (b) such that the total number of substitutions is minimized. For this tree, at least 6 substitutions are required. (d) shows a maximum parsimony tree with hypothetical sequences assigned to internal vertices. This tree requires only 5 substitutions which is the least number of substitutions required for any tree and any assignment of sequences to internal vertices.
Assume that we are given n sequences. For each rooted bifurcating tree with n leaves corresponding to the n sequences and an assignment of hypothetical sequences to the internal vertices, we can compute the minimal sets of evolutionary events that have taken place along each tree edge. Traditionally, an evolutionary event is defined as a single substitution of one nucleotide or amino acid for another. If we assign a cost to each possible substitution, we can compute a cost for each tree and assignment. The cost of a tree is then defined as the minimum cost over all possible assignments of sequences to its internal vertices. Out task is now to find the tree with the least cost. See Figure 2.5 for a complete example. A general algorithm for computing the minimal cost of a given tree was given by Sankoff [137] and Sankoff and Rousseau [138]. Finding the best tree can be done by searching in the space of trees using different heuristics and local search algorithms.
2.4. PHYLOGENY AND TREE RECONSTRUCTION
19
The criteria given above is quite general and various special cases together with specialized algorithms have been described in the literature. Other variants of parsimony exist that are not easily solved using the Sankoff algorithm, for example polymorphism parsimony [48, 51]. Distance matrix methods comprise another major class of phylogenetic methods. These methods were introduced in [25] and [58]. We can associate with each edge of a phylogenetic tree a branch length representing the total amount of evolution that has occurred between the two vertices. By summing the branch lengths of all the edges on the path between two leaves, we can compute pairwise distances between each pair of sequences or species. Distance matrix methods take as input pairwise distances between sequences and attempt to reconstruct a phylogenetic tree with branch lengths such that the distances induced by the tree is as close as possible to the given distances. In order to develop algorithms for the problem, we need to define precisely how to measure closeness between the given pairwise distances and those induced by a tree. A popular, and statistically justifiable, criterion is least squares. Let Dij be T the given distance between sequence i and j and let Dij be the distance as induced by a tree T . The best tree according to the least squares criterion is a tree T that minimizes the following expression: T 2 (Dij − Dij ) . (2.1) ij
There are efficient algorithms for determining branch lengths that minimize (2.1) given a fixed tree [63]. Finding the optimal tree according to the least squares criterion is, however, NP-hard [35]. Searching for a good tree is usually done by heuristics and local search methods. Another criterion used by distance matrix methods is that of minimum evolution. In the minimum evolution methods, the branch lengths of a tree are determined by the least squares criterion, but the optimality criterion used to choose between trees is different. Instead of choosing the tree whose induced distances is as close as possible to the given distances, the tree with the minimum total length is preferred [87, 133]. There are also other heuristic distance matrix methods that do not have an explicit optimality criterion. One of the most popular methods for tree reconstruction is called neighbor-joining (NJ) [134] and is based on the clustering algorithms popularized by Sokal and Sneath [142]. Although NJ is defined by its algorithmic description, it is related to both the minimum evolution and least squares criteria. An interesting property of NJ is that it will reconstruct the correct tree if the given distances are “sufficiently close” to the distances induced by the tree. More formally, a distance matrix D is said to be nearly additive if there is a tree T with induced distance matrix DT such that T |D − Dij |∞
0 and Pr[B] > 0. The theorem itself is in no way controversial and is valid under both the frequentist and Bayesian interpretation of probability. The controversy arises with the use of priors and computation of posterior probabilities. Assume that we have a data set D which we believe to be generated by one of n different fully specified models M1 , . . . , Mn , and that the conditional probabilities Pr[D|Mi ] are well-defined and easily computable. To choose among the models, we could for example use maximum likelihood and pick the model for which Pr[D|Mi ] is greatest. The Bayesian approach is instead to compute posterior probabilities on the set of models using Bayes’s theorem: Pr[Mi |D] =
Pr[D|Mi ] Pr[Mi ] . Pr[D]
Using the law of total probability, the denominator of the right hand side of the above equation can be written as Pr[D] =
n
Pr[D, Mi ] =
i=1
n
Pr[D|Mi ] Pr[Mi ].
i=1
The probability Pr[Mi ] is called a prior and represents our belief that model Mi is the correct model before observing the data. The probability Pr[Mi |D] is called the posterior and represents our updated belief that Mi is the correct model after having observed the data. An example of a continuous case is when we want to estimate a parameter of a model. Assume that we know that a distribution M has generated the data, but the parameter θ of M is unknown. The distribution p[θ] represents how likely different values of θ are prior to having seen the data. After data has been gathered, the posterior distribution of θ is given by Bayes’s theorem: p[θ|D] =
Pr[D|θ]p[θ] . Pr[D|θ] Pr[θ]dθ
We can then estimate θ using the posterior mean E[θ|D] = θp[θ|D]dθ, or we can b find an interval (a, b) such that, for given α, Pr[a < θ < b|D] = a p[θ|D]dθ = 1 − α. The use of priors continues to be controversial, and a thorough discussion lies outside the scope of this thesis. One objection to the use of priors is that they are not objective, but see [14] for a perspective on objectivity in Bayesian and classic statistics. Another is that it may be difficult to express one’s beliefs in terms of a probability distribution and instead one might choose a certain prior for practical reasons rather than to express one’s true opinions. And besides, humans often harbor inconsistent opinions that contradict the axioms of probability theory.
38
CHAPTER 3. COMPUTATIONAL TECHNIQUES
The good news is however that priors do not always have a strong influence on the results. It can be shown that under certain conditions, an increasing amount of data will lead to more and more similar posterior distributions [16]. Others have argued for the use of non-informative priors as being an objective choice when there is a lack of prior opinion, see for example [81]. There are often other benefits to Bayesian analysis compared to classical statistics that compensate for the use of priors. For example, in phylogenetics, MCMC techniques have enabled the use of arbitrary prior distributions, and a more efficient investigation of the state space. Bayesian methods allow us to efficiently deal with high dimensional models, and to obtain marginal distributions on the parameters of interest. Also, Bayesian methods allow us to explore posterior distributions rather than just summary statistics such as mean and variance. For an introduction to Bayesian statistics, see the excellent book by Peter Lee [95].
MCMC In Bayesian inference, we are often confronted with various integration and optimization problems. We frequently need to find a marginal distribution or compute expectations. When dealing with large dimensional spaces, analytic solutions are usually not readily available. Instead, we can obtain Monte Carlo estimates by drawing iid samples from a target distribution. These samples can then be used to approximate the density or expectation of interest. Assume for example that we draw a set of iid samples x1 , . . . , xN from a density p[x] defined on a highdimensional space X. The integral f (x)p[x]dx X
can then be approximated by the sum N 1 f (xi ). N i=1
Markov Chain Monte Carlo is a method based on Markov chains that allows us to obtain samples from non-standard distributions from which we cannot draw samples directly. To use MCMC techniques we need to be able to evaluate at least ratios of the target distribution. First, we describe MCMC on finite state spaces. Assume that the state space is S = {s1 , . . . , sn }. A Markov chain on S is a sequence of random variables X1 , X2 , . . . taking values in the state space S such that Pr[Xn |Xn−1 , . . . , X1 ] = Pr[Xn |Xn−1 ]. The chain is called homogeneous if Pr[Xn = s|Xn−1 = s ] = Pr[Xj = s|Xj−1 = s ],
3.4. BAYESIAN INFERENCE WITH MARKOV CHAIN MONTE CARLO
39
for all j, n > 1. In other words, a homogeneous Markov chain lacks memory and the distribution on the states for the next step is fully determined by the current state. A Markov chain on a finite state space can be represented by a transition matrix T , where Tij = Pr[Xn = sj |Xn−1 = si ] and j Tij = 1. Now, assume that we pick a start state randomly from a distribution represented by a row vector π = (Pr[s1 ], . . . , Pr[sn ]). The distribution on the states after one transition is given by πT . In general, the probability distribution on the states after n transitions is given by πT n . If the transition matrix is both irreducible and aperiodic, then the distribution on the states will converge to a unique invariant distribution μ irrespective of the start distribution π, i.e., πT n → μ
as n → ∞,
for any distribution π. A transition matrix is irreducible if any state is reachable from any other in a finite number of steps. A state i has period k if the length of any path returning to i is always a multiple of k. A transition matrix with a state of period k > 1 is called periodic. Otherwise, it is called aperiodic. A sufficient condition to ensure that μ is the desired distribution p from which we want to sample is the so-called detailed balance criterion: pi Tij = pj Tji . When the state space is infinite, we can instead use the Metropolis-Hastings algorithm [111, 73]. Assume that we want to sample from a distribution on a multi-dimensional state space Θ = (Θ1 , . . . , Θm ). The components of Θ can be either discrete, continuous, or a mix of discrete and continuous variables. More generally, each component can be a multi-dimensional random variable in itself. As an example, consider phylogeny where Θ could be a phylogenetic tree together with lengths associated with the edges. Instead of a transition matrix, the Markov chain is now specified as a set of proposal distributions, Rk (Θk |Θ), k = 1, . . . , m, that given the current state Θ propose a change in one of the components Θk according to some distribution. In other words, the proposed state Θ obtained from Rk differs from Θ only in the kth component: Θi = Θi , i = k. The proposed new state is then accepted with probability A(Θ, Θ ), in which case Θ becomes the new current state, or is rejected and the current state remains Θ. The component to change in each step is often picked randomly in each step. In the original Metropolis algorithm, the acceptance probability is
Pr[Θ ] A(Θ, Θ ) = min 1, , Pr[Θ] so that if the probability of the proposed state is greater than the current state, it is always accepted, and otherwise, it is accepted with probability Pr[Θ ]/ Pr[Θ]. One way to achieve detailed balance under this acceptance strategy is to ensure that Rk (Θk |Θ) = Rk (Θk |Θ ).
40
CHAPTER 3. COMPUTATIONAL TECHNIQUES
Hastings generalized the Metropolis algorithm to allow non-symmetric proposal distributions. The acceptance probability is then
Pr[Θ ]Rk (Θk |Θ ) A(Θ, Θ ) = min 1, . Pr[Θ]Rk (Θk |Θ) This again ensures detailed balance. For more on Metropolis algorithms and MCMC, and discussions on convergence rates, tests of convergence, and more, see for example [120, 98].
Chapter 4
Computational Methods and Models for Duplications and LGTs This chapter gives an overview of the different phylogenetic methods concerned with gene duplications and LGTs. Algorithms have been developed for a variety of problems, such as tree reconciliation, species tree reconstruction, and orthology analysis. Tree reconciliation is the problem of explaining the differences between a species tree and a corresponding gene tree by giving a plausible evolutionary history of the latter inside the former. Species tree reconstruction refers to problems in which an optimal species tree is sought when given a set of incongruent gene trees. Alternatively, the input can consist of sequences from different gene families, in which case substitution models are also taken into account when seeking optimal species trees. In orthology analysis, the problem is determining whether or not pairs of sequences are orthologous. Data in this case can consist of either trees, sequences, or a mix of trees and sequences. The first section of this chapter deals with the observation that certain problems in parasitology and biogeography are analogous to those of molecular evolution. The subsequent three sections give a background on previous work on the problems mentioned in the previous paragraph. A common feature of all the methods discussed is their focus on duplications and LGTs. Sections 4.5 and 4.6 discuss the work presented in Papers I, II, and III of this thesis.
4.1
Trees Within Trees
The notion of a tree structure evolving inside another tree structure has been used in at least three separate disciplines: molecular evolution, parasitology, and biogeography. In each case, questions arise about how a host is tracked by an associate. In molecular evolution, genes track organisms; in parasitology, parasites track hosts; and in biogeography, organisms track areas. The structure most widely used to depict histories of hosts and associates is that of a tree. Different historic events 41
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CHAPTER 4. COMPUTATIONAL METHODS AND MODELS FOR DUPLICATIONS AND LGTS
cause the trees of hosts and their associates to be incongruent, thus creating the need to specify exactly what those events are and where they have occurred. As it turns out, each event considered in one discipline has an analogue in the others and results in the same type of incongruity between host and associate trees. The fundamental similarity between the problems in the different disciplines was not recognized until the 1990s, although similar work in the different disciplines had been done independently. A clear exposition on this subject can be found in [129]. In molecular evolution, the events under consideration are speciations, duplications, LGTs, and losses. The corresponding events in parasitology are co-speciation, independent parasite speciation, host switching, and lineage sorting, respectively. In biogeography we have vicariance, sympatry, dispersal, and extinction. The fundamental observation here is that a single model in which an associate tree evolves inside a corresponding host tree is adequate to capture all three cases. In fact, we can find more examples where such a model can be applied; for example, the evolution of protein domains inside gene trees. In the following sections, we will formulate problems and discuss methods using terms from molecular evolution.
4.2
The Duplication-Loss Model
In the duplication-loss model, we assume that any incongruities between an organismal tree and a corresponding gene tree are due to duplications and losses. In other words, we assume that the mode of genetic transfer is strictly vertical from parent to child, although genes can be duplicated or lost and this change in genetic make-up is sometimes spread to the entire population and is fixed. Clearly, the history of a set of homologous genes represented by a gene tree is then restricted to having occurred inside the edges of a corresponding species tree. Each internal gene tree vertex corresponds to either a duplication event or a speciation event. See Figure 4.1 where a gene tree is drawn inside a species tree showing the evolutionary history of a set of genes; this is an example of tree reconciliation in which a biologically feasible explanation is provided for the disparity between the host and associate trees. In 1970, Fitch [57] made a distinction between paralogous and orthologous genes, i.e., genes whose least common ancestor in the gene tree is a duplication or speciation, respectively. Similar concepts had been developed much earlier in parasitology, see for example [28]. The development of methods for detecting pairs or groups of orthologous genes is an important step in the prediction of gene function. Traditionally, trees are taken as the data to be analyzed. A species tree is assumed given, together with a gene tree that has been constructed from sequences using methods analogous to those discussed in Section 2.4. Given a species tree and a corresponding gene tree, an important problem is determining the evolutionary history of the gene tree within the species tree and to answer questions such as which pairs of genes are orthologous and which paralogous?.
4.2. THE DUPLICATION-LOSS MODEL
43
· · ·
· ·
··
A
Figure 4.1:
·
B
·
C
·
D
·
E
Example of genes evolving inside a species tree according to the duplication-
loss model.
We will use the following notation in the discussions below. We take all trees to be rooted binary trees. The edges of a rooted tree are assumed to be directed away from the root. The subtree of a tree T rooted at the vertex u is denoted Tu . The correspondence between a species tree S and a gene tree G is given via a leaf-mapping function σ : L(G) → L(S) that maps each gene to the extant species to which it belongs. For convenience, we assume that σ is extended to map sets of gene tree leaves to the corresponding sets of species tree leaves. The function mapping a set of tree vertices to their least common ancestor is denoted lca . In 1979, Goodman et al. gave a parsimony method for tree reconciliation that maps the gene tree inside the species tree such that the number of inferred duplications and losses is minimized. This mapping is called the least common ancestor mapping, λ : V (G) → V (S), and is defined by λ(u) = lca (σ(L(Gu ))). Note that λ maps vertices of G to vertices of S. When describing a possible evolutionary history G inside S, the gene tree vertices representing speciation events are associated with the species tree vertex that corresponds to the same speciation event. A duplication vertex in G is associated with the edge of S in which the duplication occurred. The interpretation of the mapping given by λ is that if u ∈ V (G) represents a speciation, then λ(u) represents that same speciation event, and if u is a duplication, then the duplication occurred along the incoming edge of λ(u). Building on the framework provided by Goodman et al., Guigó et al. attempted to find the species tree whose reconciliation with a set of gene trees requires a minimum number of duplications [65]. Their method can be described as a heuristic local search method or hill-climbing where the neighborhood of a tree is defined by nearest neighbor interchange (NNI) operations [116]. Ma et al. proved hardness results for several species tree reconstruction methods [102], and several heuristic methods for species tree reconstruction have been developed [136, 128, 151].
44
CHAPTER 4. COMPUTATIONAL METHODS AND MODELS FOR DUPLICATIONS AND LGTS
Assuming that at least one gene tree has had a constant number of lineages in each species tree lineage, there is an FPT algorithm for reconstructing the optimal species tree [68]. Going beyond parsimony methods, probabilistic models of gene evolution for the duplication-loss model have recently been proposed. In [4, 5, 139, 155], a complete framework for computational analysis in a probabilistic setting has been developed. The model is most conveniently described as a generative model that generates a gene tree and sequences on a give species tree with times associated with the species tree edges. The model of evolution is based on the standard birth-death process [86] which generates duplications and losses along the edges of a species tree resulting in a gene tree. Sequences are then generated according to an arbitrary choice of standard substitution models. Adopting a Bayesian approach and using MCMC techniques, it is possible to compute various posterior probabilities of interest such as the probability of a gene tree given sequences, or the probability of two sequences being orthologous or paralogous. Posterior probability distributions of duplication and loss rates can also be studied. The latest development in this direction extends the model with an iid model of sequence evolution rate variation across gene tree edges [155]. Methods that simultaneously consider sequence evolution and gene tree and species tree reconciliation when identifying duplications have been termed duplication analysis. An ad hoc method for duplication analysis that also takes gene order information into account was presented in [149].
4.3
The Transfer-Loss Model
Early attempts at defining evolutionary models taking LGTs into account include the network model [147, 74, 75]. A related approach considers the subtree transfer operation on a tree in which a subtree is moved to a different location. The corresponding optimization problem is to find a minimal set of subtree transfer operations that transform one given tree to another [34, 33, 76]. Nakhleh et al. developed a heuristic for phylogenetic network reconstruction given a species tree and a set of gene trees [119]. Analogous to the case of duplications, tree reconciliation problems have been considered in settings where only transfers and losses are take into account. Variations on parsimony problems were defined and considered in [69]. Probabilistic models have also been suggested. In [82], a model was described in which sequences evolve along a network. Huelsenbeck et al. developed a Bayesian framework in the context of hosts and parasites for detecting host switches in which the data to be analyzed consists of host and parasite sequences. The model considers only the case where each host is tracked by a single parasite species so that when a host acquires a new parasite, the parasite formerly associated with the host becomes extinct. In [97], a generative model for LGT, without duplications and losses, based on a Poisson process rather than a birth-death process, was presented and used to generate synthetic data. Biological data was also compared with synthetic data
4.4. THE DUPLICATION-TRANSFER-LOSS MODEL
45
in order to infer LGT rates. In [17], Boc and Makarenkov develop a method for detecting LGTs based on distances between the sequences used to infer the species tree and gene tree. A heuristic algorithm for inferring a network using a minimum number of LGTs from a species tree and a corresponding set of gene trees was developed by Nakhleh et al. in [119]. Other, non-phylogenetic, methods for detection of LGTs include the use of atypical sequence composition, which can be used to detect recent LGTs, see for example [9].
4.4
The Duplication-Transfer-Loss Model
Based on the idea of reconciled trees, Charleston developed a computer program, Jungles [26], attempting to solve a parsimony variant of tree reconciliation that considers both duplications and transfers. Unfortunately, the presentation lacks mathematical rigor and is plagued by errors in proofs. The time complexity of the method was not analyzed, and in fact, is probably exponential. Further evidence to support this conjecture is found in Paper I of this thesis where it is shown that finding most parsimonious reconciliations that are temporally feasible is NP-hard. Although probabilistic models of gene evolution for the duplication-transferloss model (DTL-model) have been defined, see for example [32], these have not previously been used to infer transfers or to reconcile trees. A probabilistic model based on the birth-death process, as described in Paper III of this thesis, was in fact used to produce synthetic data for analysis in [67]. In [32], a similar model was suggested, but was applied only to gene family sizes. The two models differ in that the model in [32] assumes that the transfer rate is constant and independent of the number of gene lineages currently present.
4.5
DTL-scenarios
In this section, we discuss the combinatorial model and methods for tree reconciliation presented in Papers I and II. The work is a contribution along the lines of the work of Goodman et al. but for the much more complicated case when both duplications, transfers, and losses are considered. We consider as input a species tree S and a gene tree G, both rooted binary trees. The association of genes with species is given by a leaf-mapping function σ : L(G) → L(S). For convenience, we extend σ to map sets of gene tree leaves to the corresponding sets of species. Given S, G, and σ, our aim in Paper I is to find the most parsimonious reconciliation explaining the evolution of G with respect to S. Adding LGT as an evolutionary event yields a complexity that requires strict formal definitions. In order to achieve both biological and mathematical soundness in our definitions, we introduce the concept of DTL-scenarios whose associated costs are defined as the number of duplications and LGTs. We do not need to consider every biologically
CHAPTER 4. COMPUTATIONAL METHODS AND MODELS FOR DUPLICATIONS AND LGTS
46
A
Figure 4.2:
B
C
·
D
·
E
Example of a non-parsimonious reconciliation in the DTL-model.
possible reconciliation between S and G in a parsimony setting, and therefore, the definition of DTL-scenarios have been carefully crafted to ensure both biological feasibility as well as non-redundancy. Consider, for example, the evolutionary history shown in Figure 4.2. Although the example is biologically possible, there is no need to consider such histories in a parsimony setting. A thorough justification for our definition is given in Paper II. A DTL-scenario for S, G, and σ consists of a partition {Σ, Δ, Θ} of the internal vertices of G, a subset Ξ ⊂ E(G), and a function γ : V (G) → V (S). The subset Ξ consists of all the transfer edges of G. The parts Σ, Δ, and Θ correspond to the speciation, duplication, and transfer vertices of G, respectively. Finally, γ maps the gene tree into the species tree indicating where the speciations, duplications, and lateral gene transfers have occurred. Formally, A DTL-scenario for a species tree S, a gene tree G, and a leaf-mapping function σ : L(G) → L(S) is an octuple (S, G, σ, γ, Σ, Δ, Θ, Ξ), where S and G are rooted binary trees, σ : L(G) → L(S) is a leaf-mapping function, γ : V (G) → V (S) is an extension of σ, Σ, Δ, and Θ form a partition of the internal vertices of G, and Ξ ⊂ E(G) is a subset of the gene tree edges such that: (I) If u is an internal gene tree vertex with children v and w, then a) γ(u) is not a proper descendant of γ(v) or γ(w) b) At least one of γ(v) and γ(w) is a descendant of γ(u) (II) (u, v) ∈ Ξ if and only if γ(u) is incomparable to γ(v) (III) If u is an internal gene tree vertex with children v and w, then a) u ∈ Θ if and only if (u, v) ∈ Ξ or (u, w) ∈ Ξ b) u ∈ Σ only if γ(u) = lca {γ(v), γ(w)} and γ(v) and γ(w) are incomparable
4.5. DTL-SCENARIOS
47 ·
· ·
A
Figure 4.3:
·
B
· · ·
· ·
C
· · D
· · E
An example of a cyclic DTL-scenario.
c) u ∈ Δ only if γ(u) is an ancestor of lca {γ(v), γ(w)} We also need to consider the fact that some sets of LGTs can lead to temporally infeasible reconciliations, an example of which is shown in Figure 4.3. We say that a DTL-scenario is acyclic if and only if (V) There is a total order < on V (S) such that a) if (x, y) ∈ E(S), then x < y b) if (u, v), (u , v ) ∈ Ξ and v is an ancestor of v , then p(γ(u)) < γ(v ) In Paper II, we prove that the condition of acyclicity given above is both sufficient and necessary to ensure temporal feasibility. A major result in Paper I is that finding most parsimonious acyclic DTLscenarios is NP-hard. However, earlier results suggest that in most data-sets cyclicity is usually not a problem. Therefore, dropping the requirement of acyclicity we develop a polynomial-time dynamic programming (DP) algorithm as well as an FPT-algorithm for finding most parsimonious DTL-scenarios. Our algorithms are applied to biological data that have been previously analyzed in the literature with respect to LGTs. In Paper II, we extend our model by allowing arbitrary costs to be associated with duplications and LGTs and give a DP algorithm for finding minimal-cost DTLscenarios. The algorithm in Paper II constitutes a considerable improvement on the time complexity of the DP algorithm in Paper I. For any species tree and gene tree pair, there are only a finite number of DTLscenarios. The algorithms mentioned so far are able to find optimal DTL-scenarios for any given cost scheme. Due to the combinatorial nature of the problem, there are sets of cost schemes with the same set of optimal DTL-scenarios. An interesting computational problem is to partition the space of cost schemes based on the sets of optimal DTL-scenarios. This is analogous to the problem of parametric sequence
48
CHAPTER 4. COMPUTATIONAL METHODS AND MODELS FOR DUPLICATIONS AND LGTS
alignment [66]. In Paper II, we give a polynomial-time algorithm for parametric tree reconciliation. With this algorithm at our disposal, we are able to obtain the set of all DTL-scenarios that are optimal under any cost scheme. We then use this method to perform tests on synthetic data, yielding very encouraging results, that show the trade-off between sensitivity and specificity for different cost schemes.
4.6
A Comprehensive Probabilistic Model of Gene Evolution
In Paper III, we develop a probabilistic model of gene evolution with duplications, LGTs, and losses. To our knowledge, this is the first such probabilistic model that has been used for inference of duplications and LGTs. We assume that a fixed species tree is given with divergence times associated with its vertices. The model is best described as first generating a gene tree with branch lengths after which some standard substitution model can be used to generate sequences. The model uses a standard birth-death process to generate a gene tree with respect to the species tree given rates for duplications, LGTs, and losses. The resulting gene tree has times associated with its edges. We achieve a relaxed molecular clock by assuming that substitution rates on gene tree edges are iid Γdistributed variables. The rates obtained from the Γ-distribution, together with edge times, induce branch lengths on the edges of the gene tree. Finally, a substitution model is used to generate sequences. In Paper III, we use the JTT model, but any standard substitution model can be used. Many interesting computational problems can be defined based on the model described above. We provide a Bayesian framework for the analysis of sequence data using MCMC techniques. We use priors on the parameters of the model, namely the rates of duplication, LGT, and gene loss and the mean and variance of the Γ distribution. A state in our Markov chain is a triple (G, l, θ), where G is a gene tree, l is a function assigning branch lengths to the edges of G, and θ is the set of birth-death rates and the mean and variance of the Γ distribution. By using standard MCMC techniques, interesting posterior distributions can be studied. Examples include the posterior distribution on the gene tree topologies, the LGT or duplication rate, and the number of LGTs. In order to use MCMC, we need to be able to compute ratios of posterior probabilities of the form Pr[G, l, D|θ], where D is the data to be analyzed in the form of sequences (since the species tree is fixed, we omit it from our notation). We can rewrite the probability of a state in our Markov chain as Pr[G, l, θ|D] =
Pr[D|G, l] Pr[G, l|θ] Pr[θ] . Pr[D]
When computing ratios of posterior probabilities, the denominator in the above expression will cancel, and therefore, we do not need to compute Pr[D]. Pr[θ] is simply our prior distribution on the parameters, and Pr[D|G, l] can be computed
4.6. A COMPREHENSIVE PROBABILISTIC MODEL OF GENE EVOLUTION
49
according to our chosen substitution model. A major contribution of Paper III is an algorithm for computing the probability Pr[G, l|θ]. More specifically, we approximate Pr[G, l|θ] by introducing discretization points on the species tree and applying a mix of dynamic programming algorithms and techniques from numerical analysis.
Chapter 5
Modeling Cancer Progression This chapter provides a short background on cancer progression models, and Section 5.2 contains a description of the work presented in Paper IV.
5.1
Overview of Current Methods
Mathematical modeling of cancer progression started more than fifty years ago with simple, yet groundbreaking, models of tumorigenesis [122, 3, 90]. The early models all assumed that cancer is a stochastic multistep process with small transition rates. A more recent example in that direction is [83]. As noted in Section 2.6, cancer progression is an evolutionary process, and therefore, it is not surprising that methods and models from population genetics have been used extensively, see [113] for a review. In this thesis, we will follow a different line of research, which started with the introduction of Oncogenetic Trees (OTs) by Desper et al. [38]. Since Vogelstein’s path model of colon cancer, numerous narrative models for progression of diverse cancer types have been suggested, for example [80, 141, 146]. Such models are often the result of ad hoc handmade reconstructions. The introduction of OTs was an attempt at a more stringent mathematical modeling of cancer progression. An OT is a rooted tree where each vertex represents a specific genetic aberration and there is a probability associated with each edge. An OT generates a set of aberrations by first choosing a set of edges, each independently and according to its associated probability. The set of vertices reachable from the root, using only the chosen edges, is then the set of generated aberrations. In this way, an OT induces a probability distribution on the power set of all aberrations. Given cross-sectional data, i.e., sets of aberrations where each set is from a unique tumor or patient, the computational task is to find the correct OT. Desper et al. showed that computing a specific weight function on the set of all pairs of vertices and then using Edmonds’s maximum branching algorithm to obtain the topology, the correct tree will be recovered with high probability. 51
52
CHAPTER 5. MODELING CANCER PROGRESSION
One problem in OTs is that once progression stops at some vertex u, i.e., when none of the outgoing edges of u are chosen in the first step, then progression cannot reach any of the descendants of u. Biological data is almost always noisy, and in any case, real cancer progression is not tree like. The result is that usually every OT, except the OTs with a star topology, assign zero probability to some of the data, and therefore, using likelihood methods is not straightforward. Another problem is that cancer progression is best described using acyclic graphs that allow an aberration to be obtained via different pathways. In an attempt to capture more of the graph-like progression of cancers, Beerenwinkel et al. used mixtures of oncogenetic trees [12, 13, 130]. In order to assign positive probabilities to all data points in the input, the topology of the first OT was kept a star tree. For inference of mixtures, they developed an EM-like algorithm, which has not been proven to deliver locally optimal ML solutions.
5.2
Hidden-variable Oncogenetic Trees
In Paper IV, we introduce Hidden-variable Oncogenetic Trees (HOTs) and mixtures thereof (HOT-mixtures) in an attempt to remedy some of the problems with traditional OTs, while taking advantage of the simplicity of tree structures. A HOT is a tree where each vertex is associated with a pair of hidden and visible variables. The hidden variable indicates true progression while the visible variable indicates the outcome of a specific experiment, e.g., the absence or presence of a genetic aberration. The hidden and visible variables associated with a vertex u are denoted Z(u) and X(u), respectively. The values of all variables are assumed to be zero (absence) or one (presence). The distribution on the values of each variable is determined by two conditional probability distributions so that a total of four conditional distributions are associated with each vertex: Pr[X(u)|Z(u) = 0], Pr[X(u)|Z(u) = 1], Pr[Z(u)|Z(p(u)) = 0], Pr[Z(u)|Z(p(u)) = 1], where p(u) denotes the parent of u. Note that the visible variable at a vertex depends only on the hidden variable at the same vertex, and that the hidden variable only depends on the hidden variable of the parent. When generating a set of aberrations, the values of the hidden variables are determined first. This is similar to oncogenetic trees, except that the hidden variable of a vertex can receive the value one even if its parent has not. The probability Pr[Z(u) = 1|Z(p(u)) = 0], which is normally small, can be interpreted as the probability that an event associated with a later stage of progression occurs spontaneously although the stages that directly precede it have not been reached. Once the values of the hidden variables are determined, the visible variables receive their
5.2. HIDDEN-VARIABLE ONCOGENETIC TREES
53
values. The probability Pr[X(u) = 1|Z(u) = 0] is interpreted as the probability of a false positive and the probability Pr[X(u) = 0|Z(u) = 1] as a false negative. The latter can also include the probability that the progression has reached u but via a different set of events than the aberration associated with u. The set of aberrations generated are the aberrations associated with the vertices whose visible variables have value one. Paper IV also includes a description of HOT-mixtures. A HOT-mixture consists of a set of HOTs, T1 , . . . , Tn , together with a probability distribution on the same. To generate data from a mixture, we first chose a HOT according to the given probability distribution and then generate a set of aberrations from the chosen HOT. Global structural EM algorithms for inferring HOTs and HOT-mixtures constitute the major computational contributions of Paper IV.
Chapter 6
Overview of Included Articles and Manuscripts Paper I: We define a combinatorial model for the reconciliation of gene and species trees using gene duplication, lateral gene transfer, and gene loss. A reconciliation is said to be cyclic if its set of transfers are temporally infeasible. We prove that finding most parsimonious acyclic reconciliations is NP-hard. However, simulations have previously shown that in most cases the most parsimonious reconciliations are acyclic. Dropping the requirement of acyclicity, we provide efficient algorithms for construction of most parsimonious reconciliations. We also analyze a biological dataset with our tools and show that our methods work well in practice. Paper II: We continue to build on the framework provided by our model in Paper I. A thorough discussion on the soundness of our model presented in Paper I is provided. Next, we extend our model to allow arbitrary costs to be associated with duplications and lateral gene transfers and develop efficient methods for finding minimal-cost reconciliations. Analogous to parametric sequence alignment, we derive polynomial-time algorithms for parametric tree reconciliation. Tests are performed on synthetic data that show the performance of our methods. Paper III: Going beyond combinatorial methods, we define a comprehensive probabilistic model of gene evolution that incorporates a birth-death process generating duplications, lateral gene transfers, and losses, together with a substitution model with a relaxed molecular clock. To our knowledge, this is the first probabilistic model used to simultaneously infer duplications and lateral gene transfers, and is more advanced than any probabilistic model that includes LGT as an evolutionary event. We present methods based on MCMC, numerical analysis, and dynamic programming for computing various posterior distributions and probabilities, including the distribution of rates, gene 55
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CHAPTER 6. OVERVIEW OF INCLUDED ARTICLES AND MANUSCRIPTS tree topologies, and counts of lateral gene transfer events.
Paper IV: We define Hidden-variable Oncogenetic Trees (HOTs) and mixtures thereof (HOT mixtures) to capture cancer progression pathways. Vertices of a HOT represent specific genetic aberrations and a pair of hidden and visible variables are associated with each vertex. The hidden variables indicate true progression, while the visible variables indicate outcome of experiments for detection of specific aberrations. Global structural EM algorithms are presented for maximum likelihood estimation of HOTs and HOT mixtures from cross sectional data. Analysis of the performance of our methods on synthetic as well as biological data are presented.
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